Let (lambda(t, x)=lambda_{1}+lambda_{2} x). Prove that [d X_{t}=muleft(X_{t} ight) d t+sigmaleft(X_{t} ight) d W_{t}+d M_{t}] where (M)

Let \(\lambda(t, x)=\lambda_{1}+\lambda_{2} x\). Prove that

\[d X_{t}=\mu\left(X_{t}\right) d t+\sigma\left(X_{t}\right) d W_{t}+d M_{t}\]

where \(M\) is the compensated martingale of an inhomogeneous Poisson process with intensity \(\lambda\left(t, X_{t}\right)\) has a solution.